On characterization of Poisson integrals of Schrodinger operators with BMO traces

TL;DR

This paper characterizes functions in BMO spaces associated with Schrödinger operators as traces of solutions to a specific PDE satisfying a Carleson condition, extending classical BMO results.

Contribution

It extends the characterization of BMO functions as traces of PDE solutions to the setting of Schrödinger operators with potentials in reverse Hölder classes.

Findings

BMO_L(Rn) functions are traces of solutions to L'u=-u_tt+Lu=0 with Carleson condition.

The Carleson condition characterizes all L-harmonic functions with traces in BMO_L(Rn).

Extension of classical BMO characterization to Schrödinger operator context.

Abstract

Let L be a Schrodinger operator of the form L=-\Delta+V acting on L^2(Rn) where the nonnegative potential V belongs to the reverse Holder class Bq for some q>= n. Let BMO_L(Rn) denote the BMO space on Rn associated to the Schrodinger operator L. In this article we will show that a function f in BMO_L(Rn) is the trace of the solution of L'u=-u_tt+Lu=0, u(x,0)= f(x), where u satisfies a Carleson condition. Conversely, this Carleson condition characterizes all the L-harmonic functions whose traces belong to the space BMO_L(Rn). This result extends the analogous characterization founded by Fabes, Johnson and Neri for the classical BMO space of John and Nirenberg.